Migration dynamics - · PDF fileMigration dynamics M. Moretto1 and S. Vergalli2 1Department of...

Migration dynamics

M. Moretto1 and S. Vergalli2

1Department of Economics, University of Padua, Padua, Italy2Department of Economics, University of Brescia, Brescia, Italy

Received 8 September 2006; Accepted 7 August 2007;Published online 28 February 2008

� Springer-Verlag 2008

Most migration flows include observable jumps, a phenomenon that is in linewith migration irreversibility. We present a real option model where the migrationchoice depends on both the wage differential between the host country and thecountry of origin, and on the probability of full integration into the host country.The optimal migration decision of an individual consists of waiting to migrate ina (coordinated) mass of individuals. The size of the migration flow depends onthe behavioural characteristics of the ethnic groups: the more ‘‘sociable’’ they are,the larger the wave and the lower the wage differential required. The second partof the paper is devoted to calibrating the model and simulating migration flowsinto Italy over the last decade. Our calibration can replicate the migration jumpsin the short term. In particular, the calibrated model is able to project the inducedlabour demand elasticity level of the host country and the behavioural rationale ofthe migrants.

Keywords: migration; real option; labour market; network effect.

JEL Classifications: F22; J61; O15; R23.

Correspondence: Sergio Vergalli, Department of Economics, University ofBrescia, via S. Faustino 74/b, Brescia 25122, Italy (E-mail: [email protected])

Vol. 93 (2008), No. 3, pp. 223–265DOI 10.1007/s00712-007-0299-6Printed in The Netherlands

Journal of Economics

1. Introduction

Much economic research deals with mass migration inflows, observingthat migration dynamics are in general characterised by gradual waves atthe beginning of their processes, followed by suddenly increasingmigration rates (so-called ‘‘migration jumps’’ or ‘‘mass immigration’’)and then again by constant entry rates. Thus, Angrist and Kugler (2003),using descriptive statistics from the Eurostat labour force surveys for 18EU and other EEA countries, observe that the late 1980s and early1990s witnessed a ‘‘marked upturn’’. Moretti (1999), studying Italianmigration in the United States and Canada, between 1876 and 1913,highlights a sharp increase in the migration flow after 1900. Aremarkable surge in immigration was also observable in the UnitedStates (Ottaviano and Peri 2005; Peri 2006; Massey 1995), in the UK(Jackman and Savouri 1992), in France (Thierry and Rogers 2004) andin Europe (Maillat 1986).1 What could be the causes of these particulardynamics? We try to answer this question by searching for anendogenous explanation of migration jumps. We offer a model thatmerges the real option approach of investment decision applied tomigration choice and the works on migration networks into a singleframework.In the economic literature, the main variable that affects the migration

decision is the wage differential between the host country and the countryof origin (Todaro 1969; Langley 1974; Hart 1975; Borjas 1990, 1994).Nevertheless, even if the wage differential is important, it is not sufficientto totally explain migrant behaviour. Evidence seems to stress the focalrole of community networks in the migrant’s choice (Boyd 1989; Bauerand Zimmermann 1997; Winters et al. 2001; Bauer et al. 2002; Coniglio2003; Munshi 2001, 2003; Heitmueller 2003). Moretti (1999), forexample, with an alternative model to Todaro’s, found evidence that boththe timing and the destination of migration could be explained by thepresence of social networks in the host country.

1 The same evidence is found in Friedberg (2001), Hatton and Williamson(2006), Pedersen et al. (2004), and Hartog and Winkelmann (2003).

224 M. Moretto and S. Vergalli

Furthermore, the fact that the migration decision is in many cases atleast partially irreversible, is a third element that may help to explain thepresence of jumps in the migration flows. In this respect, Burda (1995),following a real option approach, implemented Sjaastad’s assumption(1962) that describes migration choice in terms of investment. Burdashowed that individuals prefer to wait before migrating, even if thepresent value of the wage differential is positive, because of theuncertainty and the sunk costs associated with migration.2 SubsequentlyKhwaja (2002) and Anam et al. (2004) developed Burda’s approach bydescribing the role of uncertainty in the migration decision. Another workthat uses real option in migration is Feist (1998), in which the authoranalyses the option value of the low-skilled workers to escape to theunofficial sector if welfare benefits come too close to the net wage in theofficial sector.Assimilating the decision of each individual to migrate to a new country

as a decision on an irreversible investment, we investigated the roleplayed by social networks to help immigrants integrate into the hostcountry, where an immigrant is completely integrated when his/hereconomic and social status is no different from the natives’ status in thehost country. We did this by considering the opportunity that eachimmigrant becomes a member of a network (a community) of homoge-neous individuals, located in the host country. The community helps theimmigrants to obtain a higher wage or improve their working conditionsif there are strong ties among the members (‘‘positive networkexternalities’’). The larger the community, the higher the number of ties,the higher the flow of information on job opportunities, and therefore thehigher the probability of integrating.Nevertheless, if the number of immigrants continues to increase, labour

competition as well as higher alienation3 among immigrants inside thecommunity may reduce their net benefits (‘‘negative network external-ities’’).

2 Investment is defined as the act of incurring an immediate cost in theexpectation of future payoff. However, when the immediate cost is sunk (at leastpartially) and there is uncertainty over future rewards, the timing of theinvestment decision becomes crucial (Dixit and Pindyck 1994, p. 3).

3 This is the case in which the members of the incumbent populationdiscontinue their attraction of immigrants (see Heitmueller 2003).

Migration dynamics 225

The struggle between these two forces is shown by an invertedU-shaped benefit function which follows directly by modelling theprobability of each immigrant being totally integrated into the hostcountry a la Bass (1969).4 The Bass model5 describes the ‘‘behaviouralrationale’’ of migration flows well by focusing on the role played bytwo kinds of immigrants: the innovators or individualists, and theimitators. The innovators are those individuals that decide to migrateindependently of the decisions of others. The imitators are thoseindividuals influenced by the number of previous migrants: they shareinformation and tend to establish a network. The weight of eachdifferent type of immigrant influences the timing of migration and thenthe size of the community.6

On the one hand, the stronger the ties among individuals, the largerthe wave. On the other hand, the presence of congestion in thecommunity and/or strong competition among workers in the hostcountry delays entry.7

Finally, we calibrate the model and simulate some migration flows intoItaly in the period 1994–2000 by using the official national statistic

4 From a theoretical point of view, an U-shaped benefit function can bederived as combination of a ‘‘herd behaviour’’ and a network effect (see Baueret al. 2002) or as an application of the theory of clubs (see Vergalli 2008).

5 The Bass model was originally built to study the diffusion of new durableproducts and largely adopted in the marketing literature.

6 The distinction between innovators and imitators is reminiscent of ‘‘theupper class theory of fashion’’ (Veblen 1924) as modeled by Matsuyama (1992).In his model individuals belong to one of two groups, respectively, withconformist and with anti-conformist preferences, and the equilibrium shows achase-flight pattern, with anti-conformist playing the role of fashion leaders andconformist playing the role of fashion followers.

7 Similar results are showed by Corneo and Jeanne (1999). They describe acontinuous-time economy populated by two types of individuals, ‘‘desirabletype’’ (natives) and ‘‘undesirable type’’ (tourists), in which an action isinterpreted as a choice of location. Their model describes the dynamics of sociallocation, defining the conditions for the take-off. In particular, they show that ifan arbitrarily small number of individuals from a socially desirable groupinnovate, a large wave of imitators will follow even when the new behaviour ismore costly than the old one.


database (ISTAT)8. The results fit the theoretical approach and replicatethe observable migration jumps.

1.1 Some supporting evidence

Table 1 shows the average growth rates of certain immigration inflow intofive European countries (Germany, Italy, The Netherlands, Sweden andthe UK) for different periods. The data for Germany were taken from theStatistisches Bundesamt (Federal Statistical Office); for the Netherlands

Table 1. Average migration growth rates

Country Inflow Growth ratePeriod 1

Growth ratePeriod 2

Growth ratePeriod 3

Germany China 1.28 1995–1999 3.21 2000–2002 2.87 2003Nigeria 1.17 1995–1999 2.09 2000–2003Syria 1.09 1995–1998 1.52 1999–2003Thailand 0.76 1995–1999 1.11 2000–2003

Italy Albania 0.11 1994–1996 0.89 1997 0.12 1998–2003China 0.10 1994–1996 0.63 1997 0.07 1998–2003Philippines 0.06 1994–1996 0.24 1997 @0.03 1998–2003Romania 0.19 1994–1996 0.32 1997–2000 0.16 2001–2003

Netherlands Angola 1.83 1996–1999 4.48 2000–2002China 1.33 1996–2000 3.24 2001–2002Sudan 1.67 1996–1997 2.43 1998–2001 1.86 2002Suriname 1.41 1996–2000 2.70 2001–2002

Sweden Chile 0.88 1981–1985 2.71 1986–1989 0.46 1990–2001Ethiopia 0.81 1981–1985 4.01 1986–1992 0.93 1993–2001Ireland 1.83 1981–1988 5.20 1989–1990 1.47 1991–2001Somalia 0.25 1981–1990 3.84 1991–1994 1.43 1995–2001

UK Ghana 1.04 1992–1998 1.72 1999–2004Pakistan 1.12 1992–1998 2.04 1999–2003 1.76 2004Somalia 2.98 1992–1999 19.98 2000–2002 9.71 2003–2004Turkey 1.25 1992–1995 3.57 1996–2004

8 ISTAT (Istituto Nazionale di Statistica) is the Official National StatisticalInstitute and its database is based on data from the Ministry of the Interior,www.istat.it.


from the Statistics Netherlands (Centraal Bureau voor de Statistiek); forSweden from the Statistics Sweden (Statistiska Centralbyran) and for theUK from the Home Office, the British government. For Italy, the datawere taken from the ISTAT database for the years between 1994 and 2000and from the Caritas report9 for the period between 2001 and 2003. Boththe ISTAT dataset and the Caritas data were supplied by the Ministry ofInternal Affairs, and were up-dated and revised by the official statisticalinstitute until 200010 (therefore, the two datasets overlap in the periodbetween 1994 and 2000).We can see that the migration process does not proceed smoothly, but it

has sudden increases in the inflow growth rates. In some periods theinflow growth rate doubles (like in Germany and Netherlands),sometimes triples (like in Netherlands and UK) and sometimes increaseseven more (Italy and Sweden or for some ethnic groups in the othercountries).Looking at the immigration reforms11 (see Boeri and Bucker 2005) in

the countries and in the periods considered, we can see that they are nothomogeneous with respect to the generosity of the welfare system for theimmigrants. In two cases (UK and Netherlands) the reforms tightened thecondition for immigration, in one case (Sweden) the policy did notsubstantially change over the years and in the last two cases (Germanyand Italy), favoured immigration. Furthermore in Germany, the reformdid not directly affect immigration. Because of the heterogeneity ofreforms as a homogeneous phenomenon it seems that, at first glance, the

9 Caritas Internationalis, however, ‘‘ is a confederation of 162 Catholic relief,development and social service organisations working in over 200 countries andterritories’’ (Caritas 2003). The edition of the Caritas Statistic Immigration Reportis part of the project ‘‘The image of Migrants in Italy, Through Media, CivilSociety and the Labour Market’’, developed in the framework of the EU/EQUALInitiative, managed by the Italian Ministry of Welfare. The project has beenpromoted by the International Organisation for Migration, Caritas of Rome andthe Archive of Immigration and involves other 19 partners, including both Italianand immigrant associations. The first Caritas Report (‘‘Dossier StatisticoImmigrazione’’) was produced by the Caritas Organisation since 1991 and ithas now become an annual report on immigration.

10 For the lack of official revised data after year 2000 at the moment of oursubmission, we have used Caritas dataset for the years between 2001 and 2003.

11 See Fondazione Rodolfo Benedetti Documentation Centre, http://www.frdb.org.


immigration reforms were not the only variable affecting migrationchoice. For this we looked for an additional endogenous explanation thatcould explain migration jumps.Finally, since we focus our analysis on Italy, in Fig. 1 we have shown

the four main foreign flows and their growth rates in Italy between 1994and 2003: Albanians, Chinese, Filipinos and Romanians.12 The migrationflows have been depleted from the two important regularisations for

-80000

-60000

-40000

-20000

0

20000

40000

60000

80000

100000

120000

140000

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-0,3

0,2

0,7

1,2

1,7

2,2

Albania China PhilippinesRomania Albania growth China growthPhilippines growth Romania growth

Fig. 1. Migration jumps

12 Description of the data:

– According to ISTAT (Istituto Nazionale di Statistica), ‘‘foreigners’’ in Italy arepersons with foreign citizenship. A child born to parents who are both foreigncitizens is considered to be a foreigner as well. A child born to an Italian and aforeign parent is considered to be an Italian citizen. Once a foreigner acquiresItalian citizenship, they are not reported in official statistics as foreigners anymore.

– Data are based on the number of valid residence permits issued to foreigners asof December 31 of each year. Children under 18 years old who are registeredon their parents’ permits are not counted.


illegal immigrants introduced in Italy in 1996 and 1998, and registratedby the ISTAT database in subsequently years.13 For the sake ofcompleteness, Fig. 2 also shows the wage differentials in the sameperiod. These were obtained from the World Bank InternationalComparison Programme database and were deflated using the Bank ofItaly14 deflator.Figure 1 shows that, for all nationalities, the migration process was not

smooth. We observed ‘‘some substantial high increases in the inflowgrowth rates’’ that we defined as ‘‘migration jumps’’. In particular, we cansee an important jump in 1997 after a certain number of yearscharacterised by low waves, as if a mass of individuals was waiting forsomething to happen before deciding to migrate. Moreover, all nation-alities showed heterogeneities in their behaviour after 1997: the Chineses

10000

11000

12000

13000

14000

15000

Wag

es

16000

17000

18000

19000

1993 1994 1995 1996 1997 1998Years

1999 2000 2001 2002 2003

Albania China Philippines Romania Italy

Fig. 2. Wage levels

13 The expectation of regularisation programs foreseen by potential immi-grants, can be interpreted as an endogenous cause for migration. Nevertheless thepolitical programs are not common knowledge. In fact in the period after 1991 inItaly a quota system was imposed on the immigration flows and, therefore wasboth extraordinary and had unpredictable results.

14 http://www.bancaditalia.it.


and the Filipinos had declining flows, whereas the Romanians had asecond jump in 2000.15

Another important aspect was that the wage differential did not seem tobe the main variable driving migration flows. In all cases (except,partially, for Romania and Albania), the jumps did not occur together witha steady rise in relative wage levels, as stressed by Moretti (1999). Then,if policy choices do not completely explain migration dynamics, why dopotential migrants wait before taking their decision to migrate? What arethey waiting for? And why do they move on mass? We try to answer allthese questions by examining whether the migration investment charac-teristics and the role of ethnic groups, can explain the migration jumpsobserved in Fig. 1. Although the phenomenon may be consistent withvarious explanations, simple arguments have to do with logistics: it takestime to decide and coordinate migration. This is consistent with theprogressive acceleration in migration flows: migration delays arisebecause it is worth waiting to decide when certain fundamentaluncertainties are resolved over time and the decision is mostlyirreversible. A logistic curve also shows the fact that learning about thehost country’s labour market takes place sequentially and stronglydepends on the role of an ethnic network in the host country.This paper is organised as follows. Section 3 presents the model and the

basic assumptions. Section 4 develops the theoretical framework thatcombines real option theory and the network effects, namely the optimalmigration strategy in the presence of positive and negative externalities.Section 5 calibrates the model and Sect. 6 makes some simulations whichconfirm the theoretical results. Finally, Sect. 7 summarises the conclu-sions.

2. The model

We assume that an individual that move to another country is completelyintegrated when his economic and social status is no different from thenative one. Nevertheless, the timing of the migrant’s integration suffersfrom a phenomenon of attrition because of the lack of information about

15 The same phenomenon is also showed for five European countries in theupdated (2007) version of Vergalli (2008) at the link http://www.sergiovergalli.it.


the host country and its labour market. We also assume that in the hostcountry, a homogeneous group of people (a community/a network) existsthat can help each immigrant to increase integration. The larger thecommunity, the closer the ties among its members and then the higher theintegration probability. The number of ties also depends on idiosyncraticcharacteristics of the immigrants, that we call ‘‘behavioural rationale’’using the Bass terminology. That is, the more ‘‘sociable’’ an individual ora group of individuals, the stronger and more ties they have.

2.1 The basic assumptions

Our main assumptions are the following:

(0) There exist two countries: the country of origin where each potentialmigrant takes decisions and the host country.

(1) At any time t a risk-neutral16 individual is free to decide to migrateto the host country discounting future benefits (the wage differentialbetween the host country and the country of origin) at the constantinterest rate q.

(2) When the migrant arrives in the host country, he/she receives only apercentage n < 1 of the host wage as first entry wage.17 So definingwio as the wage of her country of origin (where i is the country), we

are able to write the wage differential as a percentage of the wage ofthe host country:

nw� woi � ½n� wo

i =w�w ¼ /0iw:

(3) In the host country there is a community of ethnically homogeneousindividuals that helps each member to integrate with the host labourmarket (or to obtain a legal job if she is working on the illegalmarket). When the immigrant is completely integrated, he/she getsthe difference between the legal host current market wage w and thewage of the country of origin wi

o, i.e.,

w� woi � ½1� wo

i =w�w ¼ /iw:

16 See Burda (1995), Khwaja (2002), and Locher (2002) for the use of thisassumption.

17 Empirical evidence shows that this is true whether the migrant finds a legalor an illegal job (see Chiswick 1978; Borjas 1990; Massey 1987).


(4) For the sake of simplicity, we assume that the country-specificpercentages /i

0and /i (/i

0B /i) are constant over time.18

(5) Each individual enters a new country undertaking a singleirreversible investment which requires an initial sunk cost K.

(6) The size of the immigrant dn is infinitesimally small compared tothe total number of previous immigrants n.

(7) Finally, the inverse labour demand for immigrants in the hostcountry at time t is an isoelastic function of the total number ofprevious immigrants n(t):

w tð Þ ¼ h tð Þn tð Þf; ð1Þ

where h is a labour-demand-specific shock, f < 0 is the elasticityand w is the average wage of the host country.19

We introduce uncertainty into the model by assuming that:

(8) The labour-demand-specific shock h follows a Brownian motion:

dhðtÞ ¼ ahðtÞdt þ rhðtÞdW ðtÞ ð2Þ

with h (t0) = h and a, r > 0 are constant over time. The componentdW(t) is a Wiener disturbance defined as dW ðtÞ ¼ eðtÞ

ffiffiffiffi

dtp

; wheree(t) * N(0,1) is a white noise stochastic process (Cox and Miller1965).

(9) The time taken to become perfectly integrated, say s, is stochasticand depends on a distribution of probability defined as:

1� Fs tð Þ � Pr s[ tpt[ 0ð Þ ð3Þ

and its corresponding hazard rate is:20

18 We calibrate them as the loss in Purchasing Power Parity with respect tothe initial year of our dataset. See Sect. 5 below.

19 There are two implicit assumptions beyond (1). Firstly, that all incumbentimmigrants have a job and that all future immigrants seek a job. Secondly, that wrefers to labour markets that are occupied mainly by immigrants so that we canignore the role of native employees (Heitmueller 2003).

20 ps(t) is the migrant’s conditioned probability of obtaining a better job attime t ? dt, if he/she has worked at a low wage until t.


ps tð Þ � fs tð Þ1� Fs tð Þ ; ð4Þ

where fs(t) is the density function or the likelihood of being perfectlyintegrated at t.

Each immigrant decides when to enter a new country maximising his/her net benefit value defined as the expected discounted stream of wagedifferentials over the planning horizon (taken infinite for simplicity)minus the entry cost K.By (1) and Assumptions 3–8 the benefits from being completely

integrated at s are given by:21

B n sð Þ; h sð Þð Þ ¼ Es

Z

1

s

e�q t�sð Þ/wðtÞdt

8

<

:

9

=

;

� Es

Z

1

s

e�q t�sð Þ/h tð Þn tð Þfdt

8

<

:

9

=

;

; ð5Þ

where B(•) accounts for the future evolution of the number of migrantsn(t), t C s. The expectation operator Es(•) is taken with respect to therandom variables s and h (t) [and then n(t)]. Next, taking into account thebenefits the immigrant may gain before integrating, we end up with a totalbenefit value at the migration time zero as:

V n; hð Þ ¼ E0

Z

s

0

e�qt/0wðtÞdt þ e�qsB n sð Þ; h sð Þð Þ

8

<

:

9

=

;

; ð6Þ

where n(0) = n; h (0) = h. By using an indicator function J[s>t] thatassumes the value one or zero depending on whether the argument is trueor false, we can write (6) as:

21 If all immigrants face the same instantaneous probability of death k dt, wecan define q ¼ bq þ k; where bq is the market rate (Dixit and Pindyck 1994,p. 200).


V n; hð Þ ¼ E0

Z

1

0

e�qtJ½s [ t�/0hðtÞn tð Þfdt þ e�qsB n sð Þ; h sð Þð Þ

8

<

:

9

=

;

: ð7Þ

Since E[J[s>t]] = 1 @ Fs(t), we can plug (3) in (7) to obtain:

V n; hð Þ ¼ E0

Z

1

0

e�qt 1� Fs tð Þ½ �/0hðtÞn tð Þfdt

8

<

:

þZ

1

0

e�qt fs tð ÞB n tð Þ; h tð Þð Þdt

9

=

;

; ð8Þ

where the expectation is now taken only with respect to h(t) (and n(t)).If the benefit value function V(•) is known, the optimal migration policy

implies that the return from migration must be at least equal to cost K atthe entry point. In other words, we need to find the curve h*(n(t)) (i.e., thevalue of the labour demand shock) at which the n(t)th migrant isindifferent between immediate entry or waiting:22

V n tð Þ; h� ðnðtð ÞÞ½ � � K ¼ 0: ð9Þ

This is what we shall do in the next section.

2.2 The entry time s and the network effect

Before turning to the migrant’s optimal policy, we need to model theprobability of integrating (3). We have defined two different groups ofmigrants:

– Innovators: those individuals who decide to migrate independently ofthe decisions of other individuals in a social system. They are thepioneers or the individualists: their decision depends on their intrinsiccharacteristics.

– Imitators: those individuals influenced in the timing of migration bythe number of previous migrants. In particular, we mean the individualswho follow the innovators. Their particular behavioural characteristic is

22 This condition is familiar in the real option theory with the name ofmatching value condition (see Dixit and Pindyck 1994).


their sociality: they have strong ties among themselves and tend toestablish a network.23

Following Bass (1969), the probability that perfect integration occurs att, given that no integration has yet occurred, is set as a linear function ofthe size of the community, i.e.,

ps tð Þ ¼ aþ bFs tð Þ; ð10Þ

where Fs(t) stands for the number of immigrants already entered; a is thecoefficient of innovation, the influence on entry regardless of the numberof previous members; b is the coefficient of imitation, the impact ofprevious members on the probability of entry at time t. By using algebraicoperations (Bass 1969, p. 217), we get:

Fs tð Þ ¼ m� n tð Þm

; ð11Þ

and the fraction of the total immigrants integrating at time t is:

fs tð Þ ¼ aþ b� að Þm

n tð Þ � b

m2n2 tð Þ; ð12Þ

where m is the (fixed) total number of immigrants over the planninghorizon, which represents the critical ‘‘saturation’’ dimension of thecommunity.24 Finally, we get limn!m fs tð Þ ¼ 0 and fs(t) is concave iffb > 0.

23 A recent economic approach calls a similar phenomenon herd behaviour,i.e., ‘‘I will go to where I have observed others go’’ (Bauer et al. 2002).

24 By observing that the cumulative function is a logistic curve, mcorresponds to the carrying capacity defined as ‘‘the number of individuals anenvironment can support without significant negative impacts to the givenorganism and its environment’’ (Vandermeer and Goldberg 2004). It correspondsto the congestion level of the community. Since fs(t) is the likelihood of beingperfectly integrated at s, the total number of immigrants in (0, s) is (See Bass1969, p. 217):

mFsðtÞ ¼ m

Z

s

0

fs tð Þdt:


By plugging (12) and (11) into (8), we simplify (8) as:

V n; hð Þ

¼ E0

Z

1

0

e�qtnðtÞf m� nðtÞm

/0 þ½aþ b�a

m nðtÞ � bm2 nðtÞ2�

q� a/

" #

h tð Þdt

8

<

:

9

=

;

:

ð13Þ

2.3 The benefit function

Network migration theory suggests that benefit is a positive function inboth wages and network size (Massey et al. 1993). However, by (13),suppressing time for the sake of simplicity, we can write the benefitfunction per unit of time as:

p n; hð Þ � u nð Þh ð14Þ

where u nð Þ � nf m�nm /0 þ BassðnÞ

q�a /h i

and BassðnÞ � aþ b�am n� b

m2 n2� �

:

Apart from shock h, each immigrant shares the same ‘‘utility’’ u(n). Theoverall shape of u(n) is ambiguous: it depends heavily on the strugglebetween the competitive effect (i.e., more immigrants reduce wagesdepending on the magnitude of the elasticity f) and the network effect[i.e., individuals gain ‘‘utility’’ by increasing the number of fellowcountrymen which in turn increases the probability of integration via theBass function Bass(n)]. According to the relative magnitude of these twoeffects, we can observe three shapes of u(n) as in Fig. 3.Let us analyse Fig. 3 from quadrant I to quadrant III for decreasing

levels of elasticity, ceteris paribus:

quadrant I: This is the general case for a not very low level of elasticityf. A relative minimum in n@ and a relative maximum in �n exist that dividethe function into three intervals:

(1) In the interval n [ (0, n@), the competition effect prevails over thenetwork effect: a new entry reduces the benefit more than the gaincaused by cooperation among members of the community.

(2) In the interval n 2 ðn}; �nÞ the network effect prevails: the benefitincreases with n until the dimension of the network reaches level �n:


(3) In the interval n 2 ð�n;mÞ the competition effect prevails: the benefitdecreases with n until the dimension of the community hits thesaturation level m. Competition is coupled with a phenomenon ofcongestion as n moves toward m.

As shown in Fig. 3, within the interval (0, n@) a level n0 exists such thatuðn0Þ ¼ uð�nÞ: Further, for n > n0 each immigrant earns benefit lower thanu(n0) until the community size reaches the relative maximum �n: Then eachimmigrant receives a lower benefit if he/she enters with a communitypopulation n 2 n0; �nð Þ:

Since the critical level of n0 depends on the relative influence of thecompetition and network effects, for different levels of elasticity we canobserve the following:

u(n) I II

u(n)

u(n”) u(n”)

n’ n” n n n”

III

u(n)

n n

u(n)=u(n’)

n

Fig. 3. Peculiar shape of u(n)


quadrant II: Even if for low value of n competition prevails over thenetwork effect, the latter dominates any other effect as n increases. Thisimplies that for each individual, it is expedient to wait for the maximumbenefit uð�nÞ before entering.

quadrant III: Since f ? 0 implies that n0 ? 0, for very low levels ofelasticity, the benefit function simply assumes an inverse U-shape.

3. Migration dynamics

Applying Ito’s Lemma to (14) and substituting (2) to eliminate dh, we getan expression for the rate of change of p in terms of the shock and thenetwork size:

dp ¼ l nð Þpdnþ apdt þ rpdw; with p0 � u n0ð Þh0 ¼ p: ð15ÞIn (15) the first term l(n) : u0 (n)/u(n) shows the direct effect of

migration flows. Migration influences the level of benefits through itseffect on the labour market equilibrium depending on the dimension ofthe community. In particular, given any value of the shock h, moreimmigrants imply a higher or lower equilibrium level of benefitsdepending on the presence of positive l(n) > 0 or negative l(n) < 0network externalities, respectively.

3.1 Optimal migration policy for n[ �n (and <n0)

If the initial size of the community is n� �n (or n B n0), we can expectmigration to work in the following way. For any fixed n, the benefits perunit of time move according to the above stochastic process withl(n)dn = 0. If they climb to a certain level p� = u(n)h�(n), migrationbecomes feasible, the network size increases from n to n ? dn and thebenefits go downward along the function u(n). Benefits will thencontinue to move stochastically without the term l(n)dn, until anotherentry episode occurs.25 This can be summarised by the followingproposition:

25 In technical terms, the threshold p� becomes an upper reflecting barrier onthe benefit process (see Harrison 1985).


Proposition 1: If n� �nðor n� n0Þ; the optimal migration policy isdescribed by the following upward-sloping curve (Fig. 4):

h�ðnÞ � b1

b1 � 1q� að Þ K

u nð Þ ; withb1

b1 � 1[ 1; ð16Þ

where q > a and b1 > 1 is the positive root of the auxiliary quadraticequation WðbÞ ¼ 1

2 r2bðb� 1Þ þ ab� q ¼ 0:

Proof : See Leahy (1993) and the Appendix. h

In the area above the curve, it is optimal to migrate: a wave of migrantswill enter in a lump to move the benefit level immediately to the thresholdcurve. In the region below the curve the optimal policy is inaction. Theindividual waits until the stochastic process h moves it vertically to h*(n)and then again a flow of migrants will jump into the host country justenough not to cross the threshold.The ‘‘utility’’ threshold that triggers migration by individual immi-

grants is identical to that of the individual that correctly anticipates theother immigrants’ strategies. This property, discovered by Leahy(1993), has an important operative implication: the optimal migrationpolicy of each individual need not take account of the effect of rivals’entry. She/he can behave competitively as if he/she is the last toenter.26

26 In other words, when an individual decides to enter, by pretending to be thelast to migrate, he/she is ignoring two things: (1) he/she is thinking that his/herbenefit flow is given by u(n)h, with n held fixed forever. Thus, as u0(n) <0, he/sheis ignoring that future entry by other members, in response to a higher value of h,will reduce ‘‘utility’’. All other things being equal, this would make entry moreattractive for the migrant that behaves myopically. (2) He/she is unaware that theprospect of future entry by competitors reduces the option value of waiting. Thatis, pretending to be the last to migrate, the individual also believes he/she still hasa valuable option of waiting before making an irreversible decision. All thingsbeing equal, this makes the decision to enter less attractive. The two effects offseteach other, allowing the migrant to act as in isolation (see Dixit and Pindyck1994, p. 291).


3.2 Optimal migration policy for n0\n\�n

For n 2 ðn0; �nÞ; the network benefit prevails over the competitive effectand then we expect the timing of an individual’s entry is influenced by theentry decisions of others.Intuition suggests that Leahy’s result cannot be extended to cover this

case. Since there are positive externalities, the higher the number ofmembers in the community the greater the advantage in terms of benefitflow. This is evident in the case of an U-shaped benefit function (quadrantIII in Fig. 3) but it also works for the general case as uðn0Þ ¼ uð�nÞ and the‘‘utility’’ is lower in within (quadrant I in Fig. 3). Therefore, althoughentering may be profitable, it is more expensive to do so first than to enterlater on, when others have already done so. This makes the triggerp� = u(n)h�(n) no longer optimal: each migrant can do better by delayingentry.27

However, as all individuals are subject to the same labour demandstochastic shock, two equilibrium patterns are possible: either thecommunity remains locked-in at the initial size n0\n\�n; sustained by

2

2,5

3

3,5

4

4,5

The

ta (

n)

5

5,5

6

6,5

1 15 29 43 57 71 85 99 113 127 141 155 169

n183 197 211 225 239 253 267 281 295 309 323 337 351 365 379 393

Albania

Triggers, −0,02

ChinaPhilippinesRomania

Fig. 4. Optimal triggers level for f2 = @0.02

27 The decision problem involved here resembles one of war of attritionwhere each agent waits for rivals to concede (Moretto 2000).


self-fulfilling pessimistic expectations (infinite delay), or a mass ofindividuals simultaneously rushes to enter. Excluding the former,28 wecan expect entry to work in the following way: for a fixed size of thenetwork, p moves according to the process (15) with l(n)dn = 0. Ifbenefits climb to p�� = u(n)h��(n), it will trigger an entry of discrete size�n� n that raises the dimension of the community instantaneously by ajump. The exact form of the trigger h�� is given in the followingproposition.

Proposition 2: If 0� n0\n\�n; the optimal migration policy for amass of individuals �n� n is described by the following flat curve(Fig. 4):

h��ðnÞ ¼ h�ðn0Þ ¼ h�ð�nÞ � b1


u �nð Þ ; ð17Þ

Proof : See Moretto (2003, 2007) and the Appendix. h

Thus starting at n, if the initial shock is below the known trigger h�ð�nÞ;all the migrants wait until h rises to this level, and then coordinate theirentry to bring the size to the optimal level n. Working back towards n0, itis verified for every n, as long as h�(n0) is equal to h�ð�nÞ: In fact, if it wereh�ðn0Þ[ h�ð�nÞ; it could be convenient to delay entry until h�ð�nÞ; becauseof a higher obtainable benefit. Once the optimal size is reached and to theright of �n; further decision to enter proceeds as explained in the previoussection without externalities. Intuitively, starting at any n0\n\�n;Proposition 2 locates the optimal entry threshold so as to maximise thetotal benefits of the incremental number of members that enter ð�n� nÞ:The shock value h�ð�nÞ that triggers this individual’s competitive run29 isthe same threshold that justifies further marginal entry under decreasingbenefits.

28 We exclude the former by using the subgame-perfectness arguments (seeMoretto 2003, 2007).

29 The term competitive run refers to Bartolini’s definition (1993).


4. Calibration

To simulate the optimal migration policy we need values for thevariables and parameters in Eq. (14). We could then calculate (16) and(17) and then solve for n�. To perform this calibration we used themigration flows for Albanians, Filipinos, Chinese and Romanians andthe wage levels (deflated using the Bank of Italy deflator) obtained fromthe ISTAT database.30 As we show below, determining values for mostof the model’s inputs is reasonably straightforward. Estimating thecoefficients of the labour demand’s stochastic process h and of the Bassprobability of integration a and b is more complex as will be discussedbelow.

4.1 Basic inputs

The parameters to be calibrated are listed in Table 2: for the discount ratewe have used a basic level q2 = 0.03 (Nordhaus 1996) and a higher levelq1 = 0.05.31 We also add a mortality rate k = 0.001 calculated by theIstituto Superiore della Sanita32 on ISTAT data.According to Assumptions 2 and 3, the differential wage is assumed to

be a constant percentage of the wage of the host country and varieswhether the immigrant is completely integrated in the host country or not:/i and /i

0, respectively. The percentage for complete integration /i has

been calibrated considering the GDP per capita based on the PurchasingPower Parity of the initial year 1993, as listed in the InternationalComparison Programme database of the World Bank. If the immigrant isnot integrated, he/she earns only a fraction n of the wage. We havecalibrated n and then the corresponding percentage /i

0referring to the

30 For the robustness of our analysis we have calibrated our parameters byusing only the official data for the years 1994–2000.

31 Policy uncertainty regulating immigrants’ flows can also explain the choiceof two different discount rates. In particular, policy uncertainty acts as a scalefactor on the optimal threshold and it can be modelised as a poisson process. SeeVergalli (2007) and Rodick (1991).

32 http://www.iss.it.


works of Massey (1987), Borjas (1990) and Chiswick (1978).33 Theresulting /i

0and /i are shown in Table 3.

4.2 Demand volatility

The calibration of Italian immigrants’ labour demand elasticity has twoproblems: the lack of studies on Italy’s demand function for immigrants34

and the lack of work that isolates the effect of immigration inflow forforeigners and not only all (or native) workers. This problem can beovercome if we look at the EU and US work on the level of labour demandelasticity and standard deviation of the stochastic shock h. For Europeanlabour market (especially for Germany and France) some work showselasticity levels between @0.021 (Bauer 1997) and @0.24 (De New andZimmermann 1994), even if, also in these cases, some identification

Table 2. Parameters

Parameter Description Symbol Source

Discount rate q1 = 0.03q1 = 0.05

Nordhaus (1996)

Elasticity Labourdemandelasticity

f1 = @0.2f2 = @0.02

Borjas (1990), De New andZimmermann (1997),Borjas (1994), andBauer (1997)

Wage differential /i = (1 @ w�/w) /i World Bank

Wage differential /i = (n @ w�/w) /i World Bank

Entry salary Average level n Chiswick (1978)Borjas (1990)Massey (1987)

33 Massey estimates that the illegal wage is 63% of the legal wage. Borjas andChiswick show that the entry wage for each immigrant is 79 or 85% of the nativeone, respectively.

34 In this respect: Gavosto et al. (1999) show a positive elasticity for natives(+0.01) that the author justifies as a short-term effect; Venturini (1999) finds along term elasticity in the non-regular labour market between @0.01 and @0.02;Venturini (1997) calculates an elasticity level of @0.3 and @0.5 among all theworkers.


problems about immigrants elasticity, remain.35 For the US there are manypapers that try to estimate the peculiar effect of entering immigrants onlabour wages. All the US elasticity levels seem to converge towards tworepresentative values:36 @0.2 (Borjas 1990) @0.02 (Borjas 1994). On thisbasis we use the following elasticities f1 = @0.2 and f2 = @0.02, thatseem to be representative both for the US and for the EU labour market. Tocalculate an estimate of the variance of h, we used the boot-strap method,37

Table 3. /i and /i0calibration

Country /i0

/i

Albania 0.639 29/33China 0.649 8/9Philippines 0.593 5/6Romania 0.510 3/4

35 Furthermore, Peri (2005) shows an elasticity level of @0.4 among allEuropean workers; De New and Zimmermann (1994) find an elasticity level forblue collar foreigners in Germany equal to @0.24; again in Germany, Bauer(1997) has a value of @0.021; Hunt (1992) about elasticity with respect foreignersshare in occupation in France, shows a level between @0.139 and @0.08; Gangand Rivera-Batiz (1994) have a level between @0.01 and @0.11, while Garsonet al. (1987) show a level included between @0.01 and @0.04 for the Frenchlabour market.

36 Borjas (1994) reviewing the literature argues that the value of the elasticityshould be between @0.01 and @0.06. Dos Santos (2000) affirms that ‘‘from anempirical point of view,many studies attempt to estimate the impact of immigrationon wages. The elasticity of wages with respect to the number of immigrants isgenerally found to be between @0.01 and @0.02@. Garson (1987), using 1985 data,coming from ISEE, finds a level of elasticity between @0.01 and @0.04. Borjas(1990), using data from the US census of 1990, shows a level around @0.2 and in arecent paper (Borjas 2003) he obtains an elasticity around @0.33. Antonji and Card(1991), using data of the US census 1970–1980, finds a level of @0.3.

37 The bootstrap method is a computer-based method for assigning measuresof accuracy to sample estimates (Efron and Tibshirani 1994). This techniqueallows estimation of the sample distribution of almost any statistic using simplemethods (Varian 2005), like resampling with replacement from the originalsample, most often with the purpose of deriving estimates of standard errors andconfidence intervals of a population parameter like a mean, median, proportion,odds ratio, correlation coefficient or regression coefficient. In our case, by usingknown parameters (elasticity level, wage level, number of immigrants),generating errors 1000 times, we have obtained two unknown levels of variancefor each elaticity value.


obtaining two levels of variance r1 and r2 for each flow, corresponding,respectively, to f1 = @0.2 and f1 = @0.02. These values are reported inTable 4.

4.3 Bass parameters

Finally for the parameters of the Bass model (i.e., a, b, m), we employ therecursive method proposed by Bass (1969, p. 224) using the years 1996,1997 and 1998 as initial conditions.38 The results are described inTable 4. Simple observation shows that in all cases, the coefficient b isgreater than a, which guarantees the concavity of the Bass functionBass(n).

5. Results

To compare different migration inflows, we simulate the optimal triggerlevels Eqs. (16) and (17), for the four migration waves, in the case ofelasticity levels @0.02 and @0.2. Because of the difficulty of perfectlyquantifying the migration costs, we normalise K to the same arbitrary

Table 4. The Bass parameters for 1996, 1997 and 1998

Albania China Philippines Romania

r1 0.063 0.061 0.057 0.047r2 0.055 0.055 0.056 0.054b 0.973 0.850 0.648 0.828a 0.110 0.117 0.141 0.123m 0.274 0.138 0.256 0.115

38 By Considering (12) as the basic equation, we know that:nðtÞ ¼ mf tð Þ ¼ amþ ðb� aÞn tð Þ � b

m n2 tð Þ: In estimating the parameters from

discrete time series data we use the following analogue: n(t) = j ? vn(t @ 1)? zn2(t @ 1), for t = 2, 3, . . ., where n(t) immigrants at t, and nðt � 1Þ ¼

P

nðtÞcumulative immigrants through period t @ 1. Since j estimates am, v estimates(b @ a), and z estimates @(b/m): @mz = b, j/m = a. Then (b @ a) = @mz @ j/m = b, and zm2 ? vm ? j = 0, or m ¼ ð�v�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2 � 4zjp

Þ=2z; and the param-eters a, b and m are identified. See Bass (p. 219) for further details.


level for all cases.39 The principal results are shown in Figs. 4 and 5 andare displayed in Table 5.Some remarks are in order:

(1) In all ethnic groups, the wave starts when the network size, n�

m

� �

;

reaches 30 or 40% of the critical saturation level m,40 for f1 = @0.2and f2 = @0.02, respectively. Yet, the lower the elasticity level thebigger the wave, that is, as market competition increases the networkeffect and the ties among immigrants reduce and they seem to beunable to coordinate entry perfectly.

(2) The higher the elasticity, the higher the threshold level h� and thelower the migration flow. This fact depends on the sum of two effects:(i) the labour market competition, increasing with the absolute levelof f; (ii) the network effect that depends on the probability of being

Table 5. Main results

Parameters 0� n0/m n0 n�/m n� n, 1997

Albania0.05; @0.2 83.95 0.008 2,275 0.33 91,000 101,6340.05; @0.02 12.48 0.000 0 0.42 115,1500.03; @0.2 22.56 0.004 1,225 0.35 96,6000.03; @0.02 3.33 0.000 0 0.43 118,300

China0.05; @0.2 77.51 0.017 2,375 0.30 42,000 55,3520.05; @0.02 13.26 0.000 0 0.41 56,0000.03; @0.2 20.97 0.008 1,125 0.33 45,5000.03; @0.02 3.55 0.000 0 0.42 58,000

Philippines0.05; @0.2 98.81 0.000 0 0.10 25,900 93,8370.05; @0.02 16.19 0.000 0 0.36 91,7000.03; @0.2 27.58 0.066 16,975 0.23 59,8500.03; @0.02 4.38 0.000 0 0.38 96,250

Romania0.05; @0.2 85.85 0.023 2,625 0.29 34,000 44,4130.05; @0.02 15.73 0.000 0 0.40 46,2500.03; @0.2 23.17 0.012 1,375 0.32 37,0000.03; @0.02 4.21 0.000 0 0.41 47,750

39 This permits comparison of the timing and the ‘‘behavioural rationale’’among migration inflows.

40 The parameter m is described in note 23 and its values are in Table 4.


completely integrated (i.e., Bass function). The combined effectdefines the magnitude of the benefit perceived by every migrant in thehost country. On the one hand, a high number of incumbentimmigrants increases the total benefit due to the network effect. Onthe other hand, however, low wave dimensions require a high shockto trigger entry.

(3) A higher q magnifies the optimal trigger as expected.(4) The highest flows observed in the data are consistent with the

predictions of the model (i.e., n� with respect to n, 1997 in Table 5):the real wave is between the upper and the lower simulated flow inevery case studied.

(5) The higher /i0or /i, the lower the entry trigger h� as expected.

The Albanian flow is the first to start in the case of low demandelasticity and the second for high elasticity. This happens just behind theChinese flow (the second and the first, respectively), with wide jumpdimensions. Nevertheless, since the historical timing of the entries showsthat the Chinese flow is more recent than the Albanian one, the level ofelasticity on the labour market might be close to f2.

41

The timing of the migration phenomenon also depends on the particularethnic characteristics summarised in the Bass parameters: the higher theimitator’s parameter b, the earlier the migration starts. This is due to ahigh network effect that offsets labour market competition with a largerwave. Moreover, the higher the innovator’s parameter a, the lower the tiesamong immigrants and the higher the number of first entries. This canexplain the differences in behaviour among the four migration inflowsobservable in Figs. 4 and 5. In fact, the Filipinos, characterised by strongindividualist behaviour, showed a magnified first entry but a reducedjump size; vice versa, the Chinese, the Albanians and the Romanianswere characterised by higher imitator parameters and a higher wave.

5.1 Entry costs

So far, we have compared different entry triggers based on normalisedsunk costs K. This normalisation allows the Bass model to describe the

41 We should remember that, since the labour demand shock is depicted as aBrownian motion (2), the higher the threshold level, the longer the time elapsed.


migration behaviour of the flows, by defining the percentage ofinnovators and imitators in each flow, thus showing the implicit signalsthat drive the waves.We can now step back and following the theory, ‘‘quantify’’ the entry

costs faced by the four different ethnic groups by inverting (16) and (17)and evaluating them at their minimum level.42 The results are shown inTable 6 from which we derive two main points: (1) the geographicaldistance is not the focal element of sunk costs, as generally stressed in theeconomic literature. In fact, the Philippines and Romania face a K similarto Albania and China. This implies that the sunk cost faced by theimmigrant must be a wider basket of socio-economic elements; (2) it isimportant to stress that the sunk costs displayed correspond to the optimalthreshold. In all cases, since the migration occurred in the same year (i.e.,

Triggers; −0,2

15

17

19

21

23

25

27

29

31

33

35

1 15 29 43 57 71 85 99 113 127 141 155

n

The

ta (

n)

169 183 197 211 225 239 253 267 281 295 309 323 337 351 365 379 393

AlbaniaChinaPhilippinesRomania

Fig. 5. Optimal triggers level for f1 = @0.2

42 In fact, if the jump started when the trigger reached the minimum level, wecan take the value of the observed flows (see the 7th column in the Table 4) andthe level of the wage in the year of the peak and substitute these values in thefollowing equation:

K� ¼ w�b1 � 1

b1 q� að Þm� �n

m/0i þ

Bass �nð Þq� að Þ /i

� �

:


1997), the migrants entered a labour market with the same shockmagnitude. This fact meant that all immigrants gained a similar labourmarket benefit but faced different costs: for the same level of wages someethnic groups were able to face higher costs. Which element made thedifference? The answer is in the ‘‘behavioural rationale’’: high cooper-ative behaviour helped each individual to face a higher cost. Therefore,the timing of the entry should be inversely related to the sunk cost in theoptimum, i.e., Albania first, China, Romania and then the Philippines.Comparing this rank with Figs. 3 and 4, it appears that the true labourdemand elasticity should be nearer @0.02.

5.2 Saturation level

Although the simulations appear to be consistent with the ISTAT databetween 1994 and 2000, we wanted to check whether the model was alsoconsistent over time, by displaying the results of the 2004 CARITASmigration report in Table 7. According to our model, the Romaniancommunity should be near saturation level, but this fact does notcorrespond to current data by CARITAS that shows an increase inRomanian immigration waves.We suggest two explanations for this. First, our analysis uses the whole

national migration flow as a single community, and this surelyoverestimates the alienation effect. We should consider single regionalhomogeneous ethnic groups. Secondly, due to the particular method used

Table 6. Different relative entry costs

K* Albania China Philippines Romania

q1 = 0.05 1.29 1.21 1.00 1.01q2 = 0.03 1.30 1.22 1.00 1.02

Table 7. Caritas report. Number of residents per million of inhabitants, 2004

Country Number of residents

Albania 0.234China 0.100Philippines 0.074Romania 0.239


to calibrate the Bass parameters, the critical saturation level m is stronglytime-dependent. To overcome this problem we have calibrated the Basscoefficients one and two steps ahead displaying the changes in‘‘behavioural rationale’’ of the procedure.

5.2.1 Forward projection technique

In the Bass methodology, m depends on the years (initial condition) usedto calculate the parameters a and b. In particular we used 1996, 1997 and1998. We then repeated the analysis by using 1996, 1997 and 1999 andthen 1996, 1997 and 2000. Values for a, b and m are reported in Tables 8and 9, respectively.In Fig. 6, we show three curves for the Albanian triggers. theta98 is the

benchmark case calibrated with the years 1996, 1997 and 1998, theta99with 1996, 1997 and 1999 and finally theta00 with 1996, 1997 and 2000.The same method applied in Fig. 7 for the Romanian flow.Moving ahead, the last year in calibrating the Bass parameters caused a

substantial change in the shape of the entry trigger functions. In particular,in both the figures, m(t) increases from theta98 to theta00 which implies,ceteris paribus, an increase in the size of the jump. Yet, the network effectis magnified, diluting the innovators’ weight (this is why h� increases forn? 0).The higher the imitators’ coefficient, the greater the perceived saturation

dimension will be.43 Therefore, if an ethnic group has strong ties, itscommunity will probably increase more than other groups, ceterisparibus. Another effect of the increasing imitator’s coefficient with time,


t1 Albania China Philippines Romania

b 0.991 0.883 0.652 0.835a 0.097 0.103 0.137 0.108m 0.312 0.157 0.262 0.132

43 Comparing the Filipino flow to the Albanian flow we notice that theAlbanian growth rate in the saturation level is higher than the Filipino one (seeTables 7 and 8).


is that, the stronger the network effect, the lower the shock required tomigrate. This result is clearly shown in Fig. 6, but it does not emergefrom Fig. 7. The explanation of this odd result depends on the peculiar

0

5

10

15

20

25

30

Theta (n)_Albania

35

40

45

theta 98theta 99theta 00

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102

108

114

120

126

132

138

144

150

156

162

168

174

180

186

192

198

n

The

ta*

Fig. 6. Forward projection technique: Albanian threshold

0

10

20

30

40

50

60

70

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

n

The

ta*

45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

theta 98

Theta (n)_Romania

theta 99theta 00

Fig. 7. Forward projection technique: Romanian threshold


Romanian flows characterised by two jumps. In particular, moving aheadto the last year, the Bass parameters incorporate even the second jump.Finally, comparing Tables 4, 8 and 9, we can highlight how the

‘‘behavioural rationale’’ changes for the four flows: the Chinese,Albanians and Romanians become more cooperative, whereas theFilipinos seem to remain more individualists. This may explain whysome communities tend to explode and others increase at a constantrate.

6. Conclusions

This paper has tried to explain why migration flows are characterisedby observable jumps. Real option theory suggests that migration maybe delayed beyond the Marshallian trigger since the option value ofwaiting may be sufficiently positive in the face of uncertainty.Possibly, waiting may resolve uncertainty and thus enable avoidance ofthe downside risk of an irreversible investment. Burda (1995) was thefirst to use real option theory to explain slow migration rates from Eastto West Germany despite a large wage differential. Subsequent work(Khwaja 2002; Anam et al. 2004) has developed this approachdescribing the role of uncertainty in the migration decision. Recentpapers (Moretti 1998; Bauer et al. 2002) show, however, that the roleof the community is important in the migration decision. In this paper,we have shown a real option model where the choice to migratedepends on the differential wage and on the probability of beingintegrated into a host country. The corresponding integration probabil-ity is modelled following the Bass model (1969) where the ‘‘behav-ioural rationale’’ of the migration flows is shown by two kinds ofimmigrants: innovators or individualists and imitators. The weight of


t2 Albania China Philippines Romania

b 1.194 1.077 0.667 0.952a 0.073 0.086 0.122 0.059m 0.412 0.189 0.294 0.242


each different type influences the timing of migration and the size ofthe community. The closer the ties among the individuals, the higherthe dimension of the wave and the higher the entry cost faced, ceterisparibus.Furthermore, we have highlighted two opposing forces that influence

entry: on the labour market side, strong competition among workers in thehost country delays entry; at the same time, the more immigrants, thehigher the network effect that reduces the optimal threshold andanticipates entry.Simulations of some migration flows into Italy over the last twenty

years fit the theoretical approach and replicate the observable migrationjumps at least in the short-term. The model is able to project theinduced labour demand elasticity level of the host country and the‘‘behavioural rationale’’ of the migrants. Nevertheless, the use ofnational flows, as a proxy for the size of the communities, probablyoverestimates the results, suggesting future disaggregation of the ethnicflows.

Appendix A

This appendix is dedicated to proving Propositions 1 and 2 in the text. Todo this we rely on the work of Leahy (1993), Bartolini (1993), Dixit andPindyck (1994), and Moretto (2003, 2007).To determine the migrant’s optimal entry policy, the first thing to do is

to find his/her value of being perfectly integrated given each individual’soptimal future entry policy. A solution for Eq. (8) can be obtained startingwithin a time interval where no entry occurs (n, h < h�). By the typicalmethodology of real options, we obtain the general solution for Eq. (8) as(Dixit and Pindyck 1994, p. 181):

V n; hð Þ ¼ A1 nð Þhb1 þ A2 nð Þhb2 þ v n; hð Þ; ð18Þ

where 1 < b1 < q/a, b2 < 0 are, respectively, the positive and the negativeroot of the characteristic equation WðbÞ ¼ 1

2 r2bðb� 1Þþ ab� q ¼ 0;and A1, A2 are two constants to be determined.To keep V(n, h) finite as h becomes small, i.e., lim

h!0V ðn; hÞ ¼ 0; we

discard the term in the negative power of h setting A2 = 0. Moreover, the


boundary conditions also require limh!1 V ðn; hÞ � vðn; hÞf g ¼ 0; wherethe second term in the limit is the discounted present value of the benefitflows over an infinite horizon starting from h with n fixed. By Eq. (13) weget:

v n; hð Þ ¼ m� n

m

/0hnf

q� aþ aþ b� a

mn� b

m2n2

� �

/hnf

q� að Þ2: ð19Þ

Remembering that u nð Þ ¼ nf m�nm /0 þ Bass

q�a /h i

and BassðnÞ � aþ½b�am n� b

m2 n2�; the general solution of Eq. (18) becomes:

V ðn; hÞ ¼ A1ðnÞhb1 þ huðnÞq� a

: ð20Þ

It is worth noting that for a B b the function u(n), is shaped accordingto Fig. 3. Since the last term represents the value of being in thecommunity in the absence of new entry, then A1ðnÞhb1 must be thecorrection due to the new entry, therefore A1(n) must be negative. Todetermine this coefficient for each n, we need to impose suitableboundary conditions. First of all, free entry requires the (idle) migrant toexpect zero benefits on entry. Then, indicating with h�(n) the value ofthe shock h at which the n-th individual is indifferent to immediate entryor waiting for another opportunity, the condition (9) in the text(matching value condition) becomes:

V ðn; h�ðnÞÞ � A1ðnÞh�ðnÞb1 þ uðnÞh�ðnÞq� a

¼ K: ð21Þ

Secondly, the number of migrants n affects V(n, h) depending on the signof h�(n). Since hb1 is always positive, any change in n either raises orlowers the whole function V(n, h), depending on whether the coefficientA1(n) increases or decreases. Therefore, by totally differentiating Eq. (21)with respect to n we obtain:

dV ðn; h�ðnÞÞdn

¼ Vnðn; h�ðnÞÞ þ Vhðn; h�ðnÞÞdh�ðnÞdn

¼ 0 ð22Þ

¼ Vhðn; h�ðnÞÞdh�ðnÞdn

¼ 0; ð23Þ


where, (since each individual rationally forecasts the future path of newentries by competitors), Vn(n, h�(n)) = 0 (Bartolini 1993, Proposition1).44

In conjunction with the Eq. (21), the above extended smooth pastingcondition states that either each migrant exercises his/her entry optionat the level of h at which the value is tangent to the entry cost, i.e.,Vh(n, h*n)) = 0, or the optimal trigger h�(n) does not change with n.While the former means that the value function is smooth at entry andthe trigger is a continuous function of n,45 the latter case states that, ifthis condition is not satisfied, an individual would benefit frommarginally anticipating or delaying the entry decision. In particular ifVh(n, h*(n)) < 0, it means that the value of staying in the host countryis expected to increase if h falls (investing now will be expected tolead to almost certain benefits), on the contrary if Vh(n, h�(n)) > 0 itmeans that a member would expect to make losses because of adecrease in h. In both situations Eq. (23) is satisfied by imposingdh�ðnÞdn ¼ 0; therefore the same level of shock may either trigger entry by

a positive mass of migrants or lock-in the community at the initiallevel of members.46

It should be noted that using Eqs. (20), (21) and (23), it is possible tofind the optimal threshold function. The solution depends on theconcavity of u(n). As we have seen in the previous part, a genericrepresentation of u(n) distinguishes three intervals for the particular shapeof the benefit function. Let us now solve the model backwards.

A.1 Proof of Proposition 1

For the case of n B n0 or n� �n we show two things: (i) the smooth pastingcondition (23) reduces to Vh(n, h�(n)) = 0; (ii) the optimal trigger h�(n) is

44 Note that this is a generalisation of the condition in Dixit (1993, p. 35). Ifthe migrant pretends to be unique or the last entering the host country, thenu0(n) = A0(n) = 0 and the first-order condition reduces to Vh(n, h�(n)) = 0

45 Moreover, as we assumed that the individual’s size is infinitesimal, then thetrigger level h�(n) is also a continuous function in n.

46 If this condition does not hold, the expected benefit gain or loss at h�(n)would be infinite due to the infinite variation property of the stochastic process h.


equivalent to that of an individual in isolation, that is of a migrantpretending to be the last to immigrate.For (i), let us consider the value of a migrant being in the host country

starting at point (n, h < h�), and subject to the possibility of new entrieswhen h hits h�. Indicating with T the first time that h reaches the triggerh�, the optimal entry policy must then satisfy:

V ðn;hÞ¼maxh�

E0

Z

T

0

e�qt 1�FsðtÞð Þ/0hðtÞnfþfsðtÞB n;hðtÞð Þ

dt

2

4

þZ

1

T

e�qT 1�FsðtÞð Þ/0hðtÞnðtÞfþfsðtÞB nðtÞ;hðtÞð Þn o

dt

3

5

¼maxh�

E0

"

Z

T

0

e�qt m�nm

/0hðtÞnfþ½aþb�am

n� b

m2n2�/hðtÞnf

q�a

� �

dt:

þe�qTV n;h� nð Þð Þ#

;

ð24Þ

where V(n, h�(n)) represents the optimal continuation value of staying inthe host country. Because, by Eq. (21), the present value of benefits at T isK, the above value can be written as:

V ðn; hÞ ¼ maxh�

uðnÞE0

Z

T

0

e�qthðtÞdt

2

4

3

5þ KE0½e�qT�

2

4

3

5

or, after simplification (Moretto 2003, 2007):

V ðn; hÞ ¼ maxh�

uðnÞhq� a

� uðnÞh�

q� a� K

�

hh�

�b1

" #

: ð25Þ

The value of being perfectly integrated Eq. (25) is the difference betweenthe value of a migrant with a myopic strategy pretending to be the last to

have to migrate uðnÞhq�a and the value of an idle individual pretending to be


the last to migrate as expressed by uðnÞh�q�a � K

� �

hh�� b1 : To choose opti-

mally h*, the first-order condition is:

oV

oh�¼ ðb1 � 1Þ uðnÞ

q� a� b1

K

h�

� �

hh�

�b1

¼ 0 ð26Þ

and the optimal threshold function takes the form:

h�ðnÞ � b1


uðnÞ ; withb1

b1 � 1[ 1: ð27Þ

Since u(n) decreases in the interval ½�n;m�; h*(n) increases. Moreover,substituting Eq. (27) into Eq. (25) we can solve for A(n) which is negativeas required by Eq. (20):

AðnÞ ¼ � ½h�ðnÞ�1�b1

b1 q� að Þ \0: ð28Þ

Finally, substituting Eq. (28) into Eq. (25) and rearranging we obtainEq. (20):

V ðn; hÞ ¼ AðnÞhb1 þ uðnÞhq� a

� � ½h�ðnÞ�1�b1

b1 q� að Þ hb1 þ uðnÞhq� a

ð29Þ

from which it is easy to verify that Vn(n, h) = 0 within the intervalh < h�(n) and zero at the boundary.Now for (ii), let us suppose that all individuals have decided to enter at

h; with h�\h: This cannot be a (Nash) equilibrium because a singlemigrant can do better by entering at h�. In fact, the flow of benefits thateach individual is able to obtain following the policy h� is the best thatthey can do, at least till T. However, by the principle of optimality, thischoice is also optimal for the rest of the period as (24) shows: if theoptimal policy of the single migrant calls for them to be active at htomorrow, it immediately follows that the optimal policy today is to enterat h�. As (24) is a continuous function in h�, the limit as h! h� showsthat h� is a Nash equilibrium (Leahy 1993, proposition 1).If the elasticity is not too low we obtain an interval n [ (0, n0) where the

competitive effect prevails over the network effect. Therefore, with theseresults, within the interval (0, n0) the optimal threshold is still given byEq. (27) until n0. Finally, for f ? 0, n0? 0.


A.2 Proof of Proposition 2

For 0� n0\n\�n we have to show three things: (i) that an individualcannot pretend to be the last to migrate and, therefore, the optimalcompetitive trigger is no longer equivalent to that of a migrant inisolation; (ii) that the candidate policy, described in the Proposition 2,satisfies the necessary and sufficient conditions of optimality; (iii) that it isa sub-game perfect equilibrium.47

Let us assume that u(n) is U-shaped as in the quadrant III of Fig. 3. For(i) and (ii), let us begin with an idle individual that follows the optimalpolicy h�(n). Since h�(n) is decreasing in the interval n\�n: the higher thenumber of members in the community the greater their entry value. Inother words, an idle migrant would maximise his/her entry option bypretending always to be the last to migrate. In fact a migrant that pretendsto be the last to enter expects an inadmissible upward jump in benefitsfollowing the policy h�(n). To see this, consider an individual that pretendsto have been the last to enter at h = h�(n); by Eq. (19) his/her value is

simply V ðn; h�ðnÞÞ � vðn; h�ðnÞÞ ¼ uðnÞh�ðnÞq�a : Then we can see that:

V ðn; h�ðnÞÞ � limh!h�ðnÞ

V ðn; hÞ ¼ h�ðnÞb1 q� að Þ [ 0; ð30Þ

This contradicts the smooth pasting condition Vh(n, h�(n)) = 0 and thenthe optimality of h�(n).To verify that the necessary conditions are satisfied, let us calculate the

value of an (incumbent) immigrant in the host country starting at the point(n, h), that would follow a policy defined by two parameters: wait until thefirst instant T at which the process h rises to a level c > h, corresponding toan immediate increase in the community size to b > n. Making use ofEq. (24) the expected payoff V(n, h) from this policy is equal to:

V ðn; h; b; cÞ ¼ E0 uðnÞZ

T

0

e�qthtdt þ e�qTV ðb; cÞ

2

4

3

5

¼ uðnÞhq� a

� uðnÞcq� a

� V ðb; cÞ� �

hc

�b1

: ð31Þ

47 See Moretto (2003) for a conjecture of how this can be proved.


If each individual were able to choose the best moment for thecommunity’s size as well as the dimension of the jump, the first-ordercondition would be:

oV ðn; h; b; cÞoc

¼ ðb1 � 1Þ uðnÞq� a

� b1

V ðb; cÞcþ oV ðb; cÞ

oc

� �

hc

�b1

¼ 0

oV ðn; h; b; cÞob

¼ oV ðb; cÞob

hc

�b1

¼ 0:

When b and c are chosen according to the candidate policy so that b ¼ �nand c ¼ h�ð�nÞ the value function reduces to (20) and the matching valuecondition requires V(b, c) = K. These properties verify that the candidatepolicy satisfies these conditions.Let the immigrant, as in Eq. (31), wait until the first time the process h

rises to the trigger level c : h�(b), corresponding to an immediateincrease of the network size to b > n, and assume also that he/she expectsno more entry after b. Therefore the expected payoff V(b, h) from thistime onwards equals the discounted stream of benefits fixed at u(b), i.e.,by Eq. (19):

V ðb; hÞ ¼ uðbÞhq� a

: ð32Þ

Comparing Eq. (32) with Eq. (20) gives A1(b) = 0. Therefore to obtainthe constant A1(n), subject to the claim that beyond b no other immigrantswill enter, we substitute Eq. (20) into the condition Vn(n, h�(n)) = 0 to

get A01ðnÞh�ðnÞb1 þ u0ðnÞh�ðnÞ

q�a ¼ 0 resulting in:

A01ðnÞ ¼ �h�ðnÞ1�b1u0ðnÞ

q� a� �ðp

�Þ1�b1

q� au0ðnÞ

uðnÞ1�b1: ð33Þ

Integrating Eq. (33) between n and b gives:

Z

b

n

A01ðxÞdx ¼ �ðp�Þ1�b1

q� a

Z

b

n

u0ðxÞuðxÞ1�b1

dx:

Taking account of the fact that A1(b) = 0, this integral gives the constantA1(n) as:


A1ðnÞ ¼ðp�Þ1�b1

b1ðq� aÞ uðbÞb1 � uðnÞb1

h i

: ð34Þ

Substituting Eq. (34) into Eq. (20), which we rewrite to make itsdependence explicit on the end size b, gives:

V ðn; h; b; h�ðbÞÞ ¼ ðp�Þ1�b1

b1ðq� aÞ uðbÞb1 � uðnÞb1

h i

hb1 þ uðnÞhq� a

: ð35Þ

As long as u(b) > u(n) the first term in Eq. (35) is positive and it forecaststhe advantage the immigrant would experience by the entry of b @ n newimmigrants when h hits h*(b). That is, if he/she were able to choose theoptimal dimension of the jump, it would be b! �n which happens the firsttime that h reaches h�ð�nÞ: Thus, as opposed to before non-sequential entryare possible, the necessary conditions would coordinate an optimalsimultaneous entry by �n� n new immigrants. If u00(n) < 0 the necessaryconditions are also sufficient. Furthermore, substituting Eq. (35) into theextended smooth pasting condition (23) and letting b! �n; we obtain:

ðp�Þ1�b1

b1ðq� aÞ uð�nÞb1 � uðnÞb1

h i

b1h�b1 þ uðnÞ

q� a

" #

dh�

dn¼ 0: ð36Þ

The term inside square brackets is always positive (i.e., there is no value

n 2 ðn; �nÞ that makes it nil), and Eq. (36) holds with dh�

dn ¼ 0: That is, all

immigrants in the range ðn; �nÞ must enter at h ¼ h�ð�nÞ:In other words, as the stochastic process h is common knowledge, each

immigrant can foresee the benefit from the entry of others and observingthe realization of the state variable h instantaneously considers when toenter by maximizing Eq. (35). Then, with simultaneous entry, theimmigrants’ optimal strategies are easy to find: each individual enters as ifhe/she were the only person to enter but with the expectation of earningall the network benefits, i.e., h�ð�nÞ is a (symmetric) Pareto-dominant Nashequilibrium for all n\�n (see Moretto 2003, 2007). In addition, as thereaction lags are literally nonexistent, none have the incentive to deviatefrom the entry strategy h! h�ð�nÞ and b! �n given that the others do notdeviate. Finally, since h is a Markov process in levels (Harrison 1985, p.5–6), the conditional expectation (31) is in fact a function solely of thestarting states so that, at each date t > 0, the immigrant’s values resemble


those described in Eq. (35) which makes the equilibrium subgameperfect.Finally, we can to deal with the general case (quadrant I and II in

Fig. 3). If n [ (n0, n@) we need first to find a network size n� such thatu(n) = u(n�) with u0(n�) > 0 and then to perform the same policy as inEq. (35) or Eq. (36). That is, the optimal entry would be ofð�n� nÞ þ ðn � nÞ immigrants the first time that h reaches h�ð�nÞ:where:

– Parameters: are, respectively, the discount factors (i.e., q1 = 0.05;q2 = 0.03) and the elasticity levels (i.e., f1 = @0.2; f2 = @0.02);

– h�: represents the optimal trigger level at which the migration wavestarts;

– n0/m: is the critical level that ‘‘triggers’’ the network effect as apercentage of the saturation dimension m;

– n�/m: is the optimal dimension of the community in percentage of thetheoretic maximum dimension m;

– n�: is the level of the community that triggers the migration flow;– n_year (i.e., n_1997) is the empirical jump observed in our data (seeFig. 1).

Acknowledgements

We acknowledge the financial support of the University of Brescia under the 60%scheme and the two-years project, protocol N. 6413/2005. The authors wish tothank Dennis Snower, Carlo Scarpa, Tito Boeri, Paolo Buonanno and LauraPoddi for their many helpful comments and suggestions. We have also benefitedfrom the comments of Giuseppe Russo and participants of the 20th NationalConference of Labour Economics, Rome, September 22 and 23, 2005.

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